| L(s) = 1 | − 64·4-s + (−27 + 122. i)5-s + 313. i·7-s − 729·9-s + 1.06e3·11-s + 4.09e3·16-s − 1.95e3i·17-s − 6.85e3·19-s + (1.72e3 − 7.81e3i)20-s − 1.29e4i·23-s + (−1.41e4 − 6.59e3i)25-s − 2.00e4i·28-s + (−3.83e4 − 8.47e3i)35-s + 4.66e4·36-s − 7.03e4i·43-s − 6.79e4·44-s + ⋯ |
| L(s) = 1 | − 4-s + (−0.215 + 0.976i)5-s + 0.914i·7-s − 0.999·9-s + 0.797·11-s + 16-s − 0.397i·17-s − 19-s + (0.215 − 0.976i)20-s − 1.06i·23-s + (−0.906 − 0.421i)25-s − 0.914i·28-s + (−0.893 − 0.197i)35-s + 0.999·36-s − 0.884i·43-s − 0.797·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.174158 - 0.216897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174158 - 0.216897i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (27 - 122. i)T \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 2 | \( 1 + 64T^{2} \) |
| 3 | \( 1 + 729T^{2} \) |
| 7 | \( 1 - 313. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.06e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.95e3iT - 2.41e7T^{2} \) |
| 23 | \( 1 + 1.29e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 5.94e8T^{2} \) |
| 31 | \( 1 - 8.87e8T^{2} \) |
| 37 | \( 1 + 2.56e9T^{2} \) |
| 41 | \( 1 - 4.75e9T^{2} \) |
| 43 | \( 1 + 7.03e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.93e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 - 5.70e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.76e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.43e11T^{2} \) |
| 83 | \( 1 - 1.68e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.96e11T^{2} \) |
| 97 | \( 1 + 8.32e11T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.40714377311029730566851134878, −11.51465693888797487826532421757, −10.30914252991741737083960805858, −9.025452411347736902397730797180, −8.321283471745709724509240336353, −6.63171806772919136295828339271, −5.47846067914526059839591439712, −3.91407554457576303444282516296, −2.52937749526561729408169231161, −0.11488636310886792373653368724,
1.16575102145797551655946574655, 3.70400221204612739074643283617, 4.65985000252412601600183141484, 5.99335947560944753396884850568, 7.80868610697612101674873429137, 8.749127113796881097990257725744, 9.590349932276628469463582780416, 10.99336194548123138165772051105, 12.20405974351430893336139821545, 13.19927034997158456077694103824
